Properties

Label 2.3600.8t7.a.a
Dimension $2$
Group $C_8:C_2$
Conductor $3600$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $2$
Group: $C_8:C_2$
Conductor: \(3600\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 5^{2} \)
Artin stem field: Galois closure of 8.8.14580000000.1
Galois orbit size: $2$
Smallest permutation container: $C_8:C_2$
Parity: even
Determinant: 1.20.4t1.a.a
Projective image: $C_2^2$
Projective field: Galois closure of \(\Q(\sqrt{3}, \sqrt{5})\)

Defining polynomial

$f(x)$$=$ \( x^{8} - 15x^{6} + 60x^{4} - 90x^{2} + 45 \) Copy content Toggle raw display .

The roots of $f$ are computed in $\Q_{ 59 }$ to precision 9.

Roots:
$r_{ 1 }$ $=$ \( 2 + 58\cdot 59 + 18\cdot 59^{2} + 24\cdot 59^{3} + 19\cdot 59^{4} + 45\cdot 59^{5} + 28\cdot 59^{6} + 42\cdot 59^{7} + 46\cdot 59^{8} +O(59^{9})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 16 + 33\cdot 59 + 44\cdot 59^{2} + 57\cdot 59^{3} + 14\cdot 59^{4} + 52\cdot 59^{5} + 13\cdot 59^{6} + 54\cdot 59^{7} + 15\cdot 59^{8} +O(59^{9})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 18 + 2\cdot 59 + 36\cdot 59^{2} + 41\cdot 59^{3} + 17\cdot 59^{4} + 31\cdot 59^{5} + 31\cdot 59^{6} + 28\cdot 59^{7} + 14\cdot 59^{8} +O(59^{9})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 27 + 5\cdot 59 + 53\cdot 59^{2} + 6\cdot 59^{3} + 31\cdot 59^{4} + 20\cdot 59^{5} + 45\cdot 59^{6} + 24\cdot 59^{7} + 20\cdot 59^{8} +O(59^{9})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 32 + 53\cdot 59 + 5\cdot 59^{2} + 52\cdot 59^{3} + 27\cdot 59^{4} + 38\cdot 59^{5} + 13\cdot 59^{6} + 34\cdot 59^{7} + 38\cdot 59^{8} +O(59^{9})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 41 + 56\cdot 59 + 22\cdot 59^{2} + 17\cdot 59^{3} + 41\cdot 59^{4} + 27\cdot 59^{5} + 27\cdot 59^{6} + 30\cdot 59^{7} + 44\cdot 59^{8} +O(59^{9})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 43 + 25\cdot 59 + 14\cdot 59^{2} + 59^{3} + 44\cdot 59^{4} + 6\cdot 59^{5} + 45\cdot 59^{6} + 4\cdot 59^{7} + 43\cdot 59^{8} +O(59^{9})\) Copy content Toggle raw display
$r_{ 8 }$ $=$ \( 57 + 40\cdot 59^{2} + 34\cdot 59^{3} + 39\cdot 59^{4} + 13\cdot 59^{5} + 30\cdot 59^{6} + 16\cdot 59^{7} + 12\cdot 59^{8} +O(59^{9})\) Copy content Toggle raw display

Generators of the action on the roots $r_1, \ldots, r_{ 8 }$

Cycle notation
$(1,7,4,6,8,2,5,3)$
$(1,4,8,5)(2,3,7,6)$
$(1,8)(4,5)$
$(2,7)(3,6)$

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 8 }$ Character value
$1$$1$$()$$2$
$1$$2$$(1,8)(2,7)(3,6)(4,5)$$-2$
$2$$2$$(1,8)(4,5)$$0$
$1$$4$$(1,4,8,5)(2,3,7,6)$$2 \zeta_{4}$
$1$$4$$(1,5,8,4)(2,6,7,3)$$-2 \zeta_{4}$
$2$$4$$(1,4,8,5)(2,6,7,3)$$0$
$2$$8$$(1,7,4,6,8,2,5,3)$$0$
$2$$8$$(1,6,5,7,8,3,4,2)$$0$
$2$$8$$(1,2,5,6,8,7,4,3)$$0$
$2$$8$$(1,6,4,2,8,3,5,7)$$0$

The blue line marks the conjugacy class containing complex conjugation.